The concept specific depth limits for specific certifications is a very dangerous one.
First, there are no "limits" for specific certifications. Agencies have no authority to set such limits. They only issue recommendations. An agency's authority to set limits extends only to their classes.
Next, some folks would have you believe an Open Water diver has a depth limit of 60 ft. That is very dangerous. Some Open Water divers have never been deeper than 20 ft. In what way is that diver qualified to dive to 60 ft? He's not. He's really not qualified to dive deeper than 30 ft at this point. Telling him his depth limit is 60 ft is irresponsible. Other Open Water divers have made four dives to 60 ft in their class (yes, I know that would be a standards violation for PADI and one or two other agencies, but most allow it). The are qualified to dive to similar depths. A 70 ft dive for a diver with that experience is perfectly reasonable. Telling him his depth limit is 60 ft is ridiculous.
Looking to the AOW certification, take that same diver with four 20 ft dives and give him an AOW class in which he makes four more dives to 20 ft and one dive to 60 ft. Most would think because he has an AOW card, he's qualified for a 100 ft dive. He's not. IMO, he's still not qualified for a 60 ft dive. OTOH, the other Open water diver with four dives to 60 ft could make two or three dives to 70 ft, another two or three dives to 80 ft and another two or three dives to 90 ft. He should be ready for a dive to 100 ft without an AOW card.
Let's all get away from this very dangerous idea that an OW card qualifies a diver to make a 60 ft dive and that an AOW card a diver to make a 100 ft dive.
Having said that, I believe it is irresponsible to take a child of 13 much deeper than 40 ft. We simply do not know how diving might affect his growth. I strongly recommend limiting exposure to high partial pressures of nitrogen until your children have stopped growing.